Calculating Angular Velocity of a Gymnast During Trampoline Routine

Understanding Angular Velocity Calculation:

Given Data: During a trampoline routine, a gymnast is tumbling in the air at 20 rad/s in a tuck position. He then extends into a layout position and doubles his radius of gyration just before landing on the trampoline bed.

Formula Used: Conservation of angular momentum.

Calculation Steps:
1. Use the equation for conservation of angular momentum:
\[ L_1 = L_f \]
2. Calculate the initial and final angular momenta:
\[ I_1\omega_1 = I_2\omega_2 \]
3. Substitute the moment of inertia for each position:
\[ m_1r_1^2\omega_1 = m_1r_2^2\omega \]
4. Utilize the relationship of radius and angular velocity:
\[ r_1^2\omega_1 = r_2^2\omega \]
5. Given that the radius doubles (\( r_2 = 2r_1 \)):
\[ r_1^2\omega_1 = (2r_1)^2\omega \]
6. Solve for the final angular velocity:
\[ 20 = 4\omega \]
\[ \omega = 5 \, rad/s \]

Final Answer: The angular velocity of the gymnast just before landing on the trampoline bed is 5 rad/s.